WANG Qi YAO Jie-yi

(School of Applied Mathematics,Guangdong University of Technology,Guangzhou 510006,China)

Abstract The paper focuses on the stability and oscillation of numerical solutions for a kind of functional differential equations.Firstly,the conditions of numerical stability and oscillation are obtained by using the θ-methods.Secondly,we studied the preservation behavior of numerical methods for the two dynamical properties,namely under which conditions the stability and oscillation of the analytic solution can be inherited by numerical methods.Finally,some numerical examples are given.

Key words Functional differential equations;θ-methods;numerical solution;stability;oscillation

0 Introduction

As a special kind of functional differential equations,the differential equation with piecewise continuous arguments (DEPCA) has aroused lots of attention.Various properties of DEPCA have been investigated deeply.Such as convergence[16],stability[4,11],oscillation[2],periodicity[1],bifurcation[3]and asymptotic behavior[5],etc.However,all papers mentioned above deal with the properties of analytic solution of DEPCA.Nowadays,it is worth noting that numerical analysis of DEPCA be of particular interest for many scientists.Some important properties such as numerical stability[8],numerical oscillation[6,7]and numerical dissipativity[15]were investigated.In recent two papers [9,17],the authors discussed numerical approximation of DEPCA in stochastic and impulsive case,respectively.In the case of PDE,some our contributions[12,13]maybe noted.Different from above cases,in the present work,we shall study both numerical stability and oscillation for a more complicated DEPCA with scalar coefficients,and get some new results.

In this paper we consider the following DEPCA

(1)

herea,b,care all real coefficients andu0is initial condition,[·] denotes the greatest integer function.In particularly,whenc=0,the equation in Eq.(1) becomesu′(t)=au(t)+bu([t]),which is exactly the case of [8].Ifb=0,the equation in Eq.(1) becomesu′(t)=au(t)+cu(2[(t+1)/2]),which is exactly the case of [10].Thus,the results in this paper are the generalization of corresponding ones in [8] and [10].The following results for the analytical stability and oscillation of Eq.(1) will be useful for the upcoming analysis.

Theorem1[14]The analytic solution of Eq.(1) is asymptotically stable for any initial value,if one of the following cases is true

(2)

and

(3)

Theorem2[14]Assume thata≠0,then Eq.(1) is oscillatory if and only if one of the following conditions is true

(4)

1 The discrete scheme

Seth=1/m(m≥1) be stepsize,we consider the linearθ-method to Eq.(1)

(5)

and the one-legθ-method to Eq.(1)

(6)

hereθ∈[0,1],un,uh([nh]) anduh(2[(nh+1)/2]) denote approximations tou(t),u([t]) andu(2[(t+1)/2]) attn,respectively.

Letn=km+l(l=0,1,…,m-1),we defineuh(tn+ηh) asukm,uh(2[(tn+ηh+1)/2]) asu2kmaccording to [8,10],where 0≤η≤1.Thus Eqs.(5) and (6) can be reduced to the same recurrence relation

(7)

It is easily seen that Eq.(7) is equivalent to the following two cases

(1) Ifa≠0,

(8)

(9)

(2) Ifa=0,

(10)

(11)

Theorem3 Assume thatλ≠0,then Eq.(1) has the numerical solution

(12)

fora≠0,wherel=0,1,2,…,m-1 and

and

(13)

fora=0,wherel=0,1,2,…,m-1 andλ=(b+1)(b+c+1)/(1-c).

ProofAssume thatunis a solution of Eq.(7) with conditionsu2jm=d2jandu(2j-1)m=d2j-1.It follows from (9) that

(14)

From (14) and (8) withl=m-1 we have

which implies that

(15)

hence

then ifa≠0 andλ≠0 we get (12).Formula (13) can be obtained in the same way.The proof is complete.

2 Stability and oscillation of numerical solution

Lemma1un→0 asn→if and only if |λ|<1,whereλis defined in Theorem 3.

Theorem4 The numerical solution of Eq.(1) is asymptotically stable for u0,if one of the following cases is true

(16)

and

(17)

ProofAccording to Lemma 1 and Theorem 3 we know that the numerical solution of Eq.(1) is asymptotically stable if and only if

(18)

fora≠0 and

(19)

fora=0.

Ifa≠0,from (18) we have the following two inequalities hold

which are equivalent to

so we can get (16).Ifa=0,similar to the case ofa≠0,we can get (17) from (19).

From Theorem 3 we have the following corollary.

Corollary1 Assume thata≠0 then

wheredj=ujmand satisfies (15).

It is easy to check that the following two lemmas are hold.

Lemma2 Sequenceul=-aαl/(αl-1) is strictly monotonic increasing forl=0,1,…,manda≠0.

Lemma3 Assume thatb+c>-aαm/(αm-1) holds,thenb+c>-aαl/(αl-1) holds forl=0,1,…,m-1 implies thatαl+(b+c)(αl-1)/a>0.

Theorem5 Assume thata≠0,then Eq.(7) is oscillatory if and only if any of the following conditions is satisfied

(20)

ProofSufficiency.It follows from (19) that the sequence {dj}oscillates under any of the condition (20).Sinceujm=djforj=0,1,…, sounalso oscillates.

Necessity.We assume that any of the following hypotheses is satisfied

(21)

Letunbe the solution of Eq.(7),then from (15) and (21) we havedj>0 forj=0,1,2,…. By Corollary 1,Lemmas 2 and 3 we get

(1) Forn=2jm+landj=0,1,2,…

(2) Forn=(2j-1)m+landj=1,2,…

So we know thatunis a monotonous sequence forn=(2j-1)m+l.On the other hand,u(2j-1)m+m=d2jatl=m,sounhas the minimum valued2jord2j-1forn=(2j-1)m+l,namely

un≥min{d2j-1,d2j}>0.

Combining (i) with (ii),we obtainun>0 forn=km+l.This contradicts the assumption thatunoscillates.The proof is complete.

3 Preservation of stability

Definition1 The set of all triples (a,b,c) which satisfy the condition (2) is called an asymptotical stability region denoted by H.

Followed by Definition 1,the numerical asymptotical stability region can be denoted by S.

Lemma4[18]Letφ(x)=1/x-1/(ex-1),thenφ(x) is a decreasing function andφ(-)=1,φ(0)=1/2 andφ(+)=0.

Lemma5[18]For allm>|a|,

(1+a/(m-θa))m≥eaif and only if 1/2≤θ≤1 fora>0,φ(-1)≤θ≤1 fora<0;

(1+a/(m-θa))m≤eaif and only if 0≤θ≤1/2 fora<0,0≤θ≤φ(1) fora>0,

whereφ(x)=1/x-1/(ex-1).

Lemma6 For allm>M,

(1) (1+a/(m-θa))m≥eaif and only if 1/2≤θ≤1 fora>0,φ(a/M)≤θ≤1 fora<0;

(2) (1+a/(m-θa))m≤eaif and only if 0≤θ≤1/2fora<0,0≤θ≤φ(a/M) fora>0,

whereφ(x)=1/x-1/(ex-1).

Proof(i) From (1+a/(m-θa))m≥eawe haveθ≥m/a-1/(ea/m-1).So for allm>M,in view of Lemma 4 we obtain 1/2≤θ≤1fora>0,φ(a/M)≤θ≤1fora<0.The case of (ii) can be proved in the same way.

Lemma7 Assume that inequalityp

(1) 1/2≤θ≤1 or 0≤θ≤φ(1) form≥Manda>0;

(2)φ(-1)≤θ≤1 or 0≤θ≤1/2 form≥Manda<0,

whereφ(x)=1/x-1/(ex-1).

ProofFor alla≠0,there exists aM0>0,whenm>M0,the range ofαmhas the following two cases

ea≤αm

Letε=min{q-ea,ea-p},ifea≤αm

We will investigate which condition leads toH?S.For convenience,we divide the regionHinto five parts

H0={(0,b,c)∈H:a=0},H1={(a,b,c)∈HH0:a>0,(a+b+c)(a+b-c)>0},

H4={(a,b,c)∈HH0:a<0,(a+b+c)(a+b-c)<0}.

In the similar way,we denote

S0={(0,b,c)∈S:a=0},S1={(a,b,c)∈SS0:a>0,(a+b+c)(a+b-c)>0},

S4={(a,b,c)∈SS0:a<0,(a+b+c)(a+b-c)<0}.

Hi∩Hj=?,Si∩Sj=?,Hi∩Sj=?,i≠j,i,j=0,1,2,3,4.

Therefore,we can conclude thatH?Sis equivalent toHi?Si,i=0,1,2,3,4.

Ifa>0,(2) becomes

(22)

(16) yields

(23)

Ifa<0,(2) turns into

(24)

(16) gives

(25)

Theorem6 The stability regions have the following five relationships

H0?S0if and only if 0≤θ≤1;H1?S1if and only if 0≤θ≤φ(1);H2?S2if1/2≤θ≤1 or 0≤θ≤φ(1);H3?S3if and only ifφ(-1)≤θ≤1;H4?S4ifφ(-1)≤θ≤1 or 0≤θ≤1/2,whereφ(x)=1/x-1/(ex-1).

Proof(i) Noticing that (3) and (17) are the same in form,soH0?S0holds for allθwith 0≤θ≤1.

(ii) By the notation ofH1andS1,(22) yields

(23)can be changed into

Therefore,H1?S1if and only ifαm≤ea,so by Lemma 5 we haveH1?S1if and only if 0≤θ≤φ(1).

(iii) By the notation ofH2andS2,(22) becomes

(23) gives

(iv) and (v) can be proved in the similar way.

4 Preservation of oscillation

Definition2 We call the θ-methods preserve oscillation of Eq.(1) if Eq.(1) oscillates,which implies that there is anh0such that Eq.(7) oscillates forh

By some easy inductions we have the next lemma.

Lemma8 For allm>|a| andθ∈[0,1],

(1) -a-1<-aαm/(αm-1)<-aand 00;

(2) -1<-aαm/(αm-1)<0 and -a

The following lemma can be naturally obtained from Theorem 2.

Lemma9 Eq.(1) is oscillatory if any of the following conditions is satisfied

(1)b+c<-aea/(ea-1),b≥-aandc>a/(ea-1) fora>0;

(2)b+c<-aea/(ea-1),b≥0 andc>a/(ea-1) for -ln2≤a<0;

(3)b+c<-aea/(ea-1),b≥0 andc>a/(ea-1),b+c>0,b<-aea/(ea-1) andc≤-afora<-ln2.

By Theorem 5,Lemmas 8 and 9 the following corollary is obtained.

Corollary2 For allm>|a|,under the condition of Lemma 9,if

holds,then the numerical solutions inherit oscillation of the analytic solutions of Eq.(1).

So we have the first result for preservation of oscillation.

Theorem7 The numerical solutions inherit oscillation of the analytic solutions of Eq.(1) if one of the following conditions holds

(1) 1/2≤θ≤1 fora>0;

(2) 0≤θ≤1/2 fora<0.

ProofFrom Corollary 2 we obtain that the numerical solutions inherit oscillation of the analytic solutions of Eq.(1) ifαm≥eafora>0 andαm≤eafora<0.Then by Lemma 5 the proof is finished.

Furthermore,from Theorem 2 we can easily obtain the following lemma.

Lemma10 Eq.(1) is oscillatory if any of the following conditions is satisfied

(1)b+c<-a-1,b>-aea/(ea-1) andc≥1 fora>a1;

(2)b+c<-a-1,b>-aea/(ea-1) andc≥1,b+c>-aea/(ea-1),b≤-a-1,c

(3)b+c≤-1,b>-aea/(ea-1) andc≥-a+1,b+c>-aea/(ea-1),b≤-1,c

wherea1is the positive root of equationea-2a-1=0.

By Theorem 5,Lemmas 8 and 10 the following corollary is got.

Corollary3 For allm>|a|,under the condition of Lemma 10,if

holds,then the numerical solutions inherit oscillation of the analytic solutions of Eq.(1).

So we have the second result for preservation of oscillation.

Theorem8 The numerical solutions inherit oscillation of the analytic solutions of Eq.(1) if one of the following conditions holds (1) 0≤θ≤φ(1) fora>0;(2)φ(-1)≤θ≤1 fora<0,whereφ(x)=1/x-1/(ex-1).

ProofFrom Corollary 3 we obtain that the numerical solutions inherit oscillation of the analytic solutions of Eq.(1) ifαm≤eafora>0 andαm≥eafora<0.Then by Lemma 5 the proof is completed.

andM=max{|a|,M*} such that

so the third result for preservation of oscillation is as follows.

Theorem9 The numerical solutions inherit oscillation of the analytic solutions of Eq.(1) if one of the following conditions holds (1) 0≤θ≤φ(1)or 1/2≤θ≤1 fora>0 andm≥M;(2)φ(-1)≤θ≤1 or 0≤θ≤1/2 fora<0 andm≥M,whereφ(x)=1/x-1/(ex-1).

ProofIt is easy to know that the range ofa/(αm-1) has two cases for allm≥M,a/(αm-1)≤a/(ea-1) and (ii)a/(αm-1)≥a/(ea-1).

The first case implies thatαm≥eafora>0 andαm≤eafora<0,then by Lemmas 5,6 we have that the numerical solutions inherit oscillation of Eq.(1) if 1/2≤θ≤1 fora>0 and 0≤θ≤1/2 fora<0.

The second case can be obtained similarly.The proof is complete.

Remark1 It is easy to see thatun=u(tn) fora=0,so the preservation of oscillation of the θ-methods is obvious whena=0.

5 Numerical simulations

We propose some examples to test the above main results.

Consider the equation

(26)

Letm=100 andθ=0.5,it is can be seen that the condition (16) is satisfied fora=1,b=1,c=-3.In Figure 1,we draw the figure of the numerical solution of Eq.(26),from this figure we know that the numerical solution of Eq.(26) is asymptotically stable,which is in agreement with Theorem 4.

For the equation

(27)

We can test thata=1,b=-1,c=-3,m=100 andθ=0.4 satisfy the third condition in (20).In Figure 2,we draw the figure of the numerical solution of Eq.(27),we can see that the numerical solution of Eq.(27) is oscillatory,which coincides with Theorem 5.

We consider the equation

(28)

it is easy to see thata=1,b=1.5,c=-3,m=100>M=2 andθ=0.4 satisfy (iii) in Theorem 6.We gave the analytic solutions and the numerical solutions of Eq.(28) in Figure 3,we can easily see that the numerical solutions inherit the stability of analytic solutions of Eq.(28),which in accordance with Theorem 6.

Consider the equation

(29)

Letm=100 andθ=0.7,it can be checked thata=-1,b=-1,c=1 satisfy (iv) in Theorem 6.In Figure 4,we draw the figures of the analytic solution and the numerical solution of Eq.(29),respectively,we can see that theθ-methods preserve the stability of Eq.(29),which is in agreement with Theorem 6.

For the equation

(30)

it is not difficult to see thata=-1,b=-0.8,c=1 satisfy (iii) in Lemma 9.Letm=100 andθ=0.4.We gave the analytic solutions and the numerical solutions of Eq.(30) in Figure 5.It shows that the numerical solutions inherit the oscillation of analytic solutions of Eq.(30),which coincides with Theorem 7.

Furthermore,for the equation

(31)

We can verify that the coefficientsa=1,b=1,c=-5 satisfy (iii) in Lemma 10.Letm=100>M=2 andθ=0.8,in Figure 6,we draw the figures of the analytic solution and the numerical solution of Eq.(31),respectively,we can see that theθ-methods preserve the oscillation of Eq.(31),which is in agreement with Theorem 9.

The other results can be tested in the same way.All numerical results show good agreement with our theoretical results.

6 Conclusions

In this paper,we consider the numerical properties ofθ-methods for a special kind of functional differential equations.Some conditions for the stability and oscillation of the numerical solution are given.The conditions that theθ-methods preserve the stability and oscillations of the analytic solutions are obtained.The numerical examples show that theθ-methods are suitable and effective for solving this kind of equation.We will consider the multidimensional case and the linear multistep methods in our further work.